Vacuum Question

A sealed container of water in a vacuum chamber. Since water doesn't expand like air, I assume that there would be no extra pressure within the sealed container once the air was evacuated around it. Since there would be no, or very little air surrounding the container, and since heat requires a medium to dissipate, would the temperature of the water inside the container stay virtually constant or close to the temperature it was when placed into the vacuum chamber? If the vacuum chamber was placed in a sub-zero environment would the water still stay around its original temperature?

Also, while on the vacuum topic, since the air in a sealed balloon will expand and blow up large in a vacuum, what would happen if that sealed balloon with a small amount of air would be placed into the above example's sealed container of water in a vacuum chamber? Would there be no extra pressure inside the container and therefore the balloon will stay its same size?

For your second question, since water is effectively incompressible, the balloon immersed in water won't expand. The pressure will change as you change the temperature, but the volume will stay the same.